Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

TL;DR

This paper investigates the maximum size of a random Dirichlet polynomial with Steinhaus coefficients, establishing bounds that grow exponentially with a fractional power of the logarithm of N.

Contribution

It provides probabilistic bounds for the extremal size of Dirichlet polynomials with random multiplicative coefficients over various ranges of t.

Findings

Upper and lower bounds on the maximum of D_N(t) with high probability.

The bounds grow roughly as exponential of a fractional power of log N.

Results hold uniformly for t in a range up to N^C.

Abstract

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$